Calculating instrument



y 5, 1953 A. F. WINTERFELDT CALCULATING INSTRUMENT 3 Sheets-Sheet 1 Filed May 12, 1952 INVENTOR. Arthur E W'interfeldt BY P 4 i4 i /W y 5, 1953 A. F. WINTERFELDT CALCULATING INSTRUMENT I 3 Sheets-Sheet 2 Filed May 12, 1952 N INVENTOR. Art/turf." Winlerfeldt y 5, 1953 A. F. WINTERFELDT 2,637,499

CALCULATING INSTRUMENT" Filed May 12, 1952 3 Sheets-Sheet 5 IN V EN TOR.

firilzurli'm'nterfieldt $4.477 Maw/W Patented y 5, 1953 a,

UNITED STATES PATENT OFFICE CALCULATING INSTRUMENT Arthur F. Winterfeldt, Montgomery County, Md.

Application May12, 1952, Serial N 0. 287,412

Claims. (Cl. 23588) (Granted under 1 The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor.

This invention relates to calculators, and more particularly to a calculating instrument which is utilized to solve a number of problems involved in the layout of animation motion pictures.

Many peculiar problems arise in the production of animation motion pictures. Generally, an animation motion picture is produced by photographing a series of separate pictures on a continuous length of film. The pictures are generally drawn by hand. Motion in the objects is produced by having the objects positioned somewhat differently in each successive picture there being sixteen separate photographs or frames per foot of film. When the continuous roll of film is developed and displayed through a motion picture projector, the different pictures are seen faster than the human eye can detect and therefore the objects appear to move.

In the art, means have been provided for causing movement of objects other than by the drawing of separate pictures, particularly straight line movement. This is accomplished by mounting a background drawing over a source of light and providing a pair of parallel sliding bars, known in the art as pan bars, on opposite sides of the drawing. That is, one bar may be mounted adjacent and parallel to the top edge of the drawing and the other bar mounted adjacent the bottom edge. The object to be moved is drawn on a transparent sheet of Celluloid which is adapted to be secured to the pan bars. This transparent sheet of Celluloid is known in the art as a pan cel. Therefore, the object which is to move is placed in its initial position and a photograph is made. The pan bars are then moved to position the object in a new position and a second photograph is taken. This process is repeated until the object is in its final position for the particular movement desired. In the layout stage, it is necessary to compute how far to move the pan bar in hundredths of an inch each time a sep-' arate photograph is made in order that the object will move a certain distance in inches over the background during the filming of a certain number of feet of film. The mental calculation of this amount of movement is normally carried out by the animator himself and is both time consuming and often erroneous. Approximately 25% of the animators time'is required for calculations. Further, should an error in the calcu-' Title 35, U. S. Code (1952),

sec. 266) costly and tedious camera work done for nothin and perhaps make it necessary to revise and re-' draw the layout and art work.

It is an object, therefore, of the present invention, to provide a calculating instrument to solve the above mentioned problem which eliminates error and saves many man hours work, it being only necessary to make two settings based on the known information and reading the answer.

Another problem involved in animated motion picture production is in the determination of the length of pan cel to accomplish the amount of movement which will be necessary. An analogous problem is the determination of the length of panoramic background so that the object can begin its movement at one point in the drawing and end its movement at another point. The animator knows the number of feet of film in which the movement must be made. For example, this may be determined by the length of narration which is to be used during the movement. He also knows approximately the rate of movement necessary to create a smooth moving effect. To have to convert the number of feet of film to the number of separate pictures or frames and multiply this by the rate of movement in hundredths of an inch is also time consuming and often results in erroneous calculations. The many times the calculations must be made during the production of a film make it imperative that a quick and eflicient means be available for the calculations. This is emphasized by the fact that by their very nature animated motion pictures are very expensive productions.

It is another object of the present invention to provide a calculating instrument which solves the problem mentioned in the paragraph above by merely manipulating the parts of the instrument until the known information has been placed in the instrument and then reading the answer.

Under special'circumsta-nces it may be desired to makethe straight line movement non-parallel to the top or bottom edges of the panoramic background. For example, it may be desiredto have an aircraft dive on a ship which is drawn on the background. This makes it necessary to move the pan bars until they parallel the dive angle of the aircraft. Under these circumstances, how- I ever, the panbars will cut opposite comers of lations occur, it results in time lost on production, x

the panoramic background drawing. Since the camera is originally positioned to take in the entire field of the drawing, it will also take in portions of the pan bars. Therefore, the camera must be repositioned to take in a smaller'fleld long side, and is the maximum field for the cam.-

era. This same camera may be moved closer to the drawing and have a minimum field which is proportionally smaller, which may begiven .a

smaller numerical value. For example, this may be known as a four field wherein the length of the longest sides of this minimum field .is four The camera support may becalibrated 1 so that the operator can tell when the "camera? "and I5, respectively.

inches.

aperture is the correct distance from the drawing for these two fields and for all fields therebetween. Therefore, when the camera is set for a twelve field and the pan bars have beentilted a certain number of degrees to provide the aircraft diving effect as aforementioned, the camera must be loweredso that the aperture takes inia maximum s-ize fiel'd which does not include the an bar's. Ordinarily the problem is solved by experimenting with different camera positions until a correc't position is found.

It is another object of this invention to pro Fig. 6 is a sectional view taken on lines 6-45 of Fig. 3;

Fig. 7 is an enlarged view of the broken away portion of Fig. 1; and

Fig. 8 is an enlarged view of the broken away portion 0f '2. J .S'i-milar reference characters represent similar parts throughout the specification.

.As seen in Figs. 1, 2 and 3, the instrument comprises a pair of spaced apart rectangular frame members It and H having spacing members Hand 13 between the ends thereof. The

frame members may be fused or otherwise securedto the spacing members. The frame memranged thereon -:on each side thereof, secon'd'a'nd third circular discs mounted on "each side or said first mentioned disc; and a frame means sur roundings'aid disc's abo'ut a-diameter thereof said frame means having a securing means "which extends therethrough and through central aper tures 'in said discs for rotatab'ly mounting said discs, .'j

'A further object or the invention is to provide an instrument :as aforementioned wherein the frame means is transparent with a hairline extending the length thereof with which infoimation printed on said 'discs'is adapted to be oriented, and a plurality of spirally arranged win-- dows :on the second and third discs which permit the reading 'of' a :COIIG'Ct answer .from the radially extending columns "on the first disc for a a given orientation of the second and third discs. Other objects \and many of the attendant advantages (of this invention will be readily ap preoiated as the same becomes .better understood by reference to the followi-ng detailed de scription when considered in connection with the accompanying drawings wherein: a 1

.E'ig. .1 is :a plan view of a "preferred embodiment of :the calculating instrument with aa portion thereof broken away to show further details; v 1 Fig. 2 is a .plan view i'of a modified :form :of the instrument also havingwparts thereof broken away to provide greater clarity; vliig. 3 is a sectionalyiew taken on lines 3 -3,

*Fig. 4 is .a sectional view taken :aon lines 4 4- of Fi 1 :Eigmh is a sectional view :taken' on'lines:?5+:-'5 b'f ili ig fl; I 1 5- bers are constructed preferably of a transparent material such as plastic and include the longitudinally extending centerlines or hairlines l4 A first mentioned circular disc I6 is mounted between the two frame members with the periphery of the disc closely adjacent the inside edge of the spacing members. A :second mentioned circular disc H of lesser diameter than disc I6 is mounted adjacent one side of disc I-6 and frame member it. A third mentioned circular disc 18. also of lesser diameter than disc i6, is mounted adjacent the other side of disc l6 and frame member 1 I. All three discs :and

the two frame members are provided with central apertures and a pivot pin l9 extends therethrough to provide ;a rotatable mounting for the discs. As shown in Fig. 6 pin t9 preferably is in the forlm of an internally threadedsleeve 2i! having :a flange J2! atone end thereof. Ascrew .22 is adapted to be received .by the sleeve, there be.- ingr a .head 23 .on one .end of the screw. Seats are provided in members H and 12 for the reception of ,fiange I21 and .head .23, respectively.

The peripheryofcircu-lar .disc i5 is marked on each side thereof with fifty short radially extending lines which are spaced equal distances from each other. Each line is marked with an identifying numeral in a series beginning with and extending through 25, each succeeding line in a clockwise direction from the beginning on one side, as seen infFig. 7,, and a counterclockwise direction .f-rom the beginning on the other side, as seen in Fig. 8, being givena value larger by A2 than the preceding numeral. These numerals-represent feet of film. On the side .of disc 16 adjacent disc lil, asseen in Figs. 1 and .7, these numerals :on the periphery form a scale .24.

. A column of numerals illustrated by the reference-character 26 extends from the center of the disc ,to each of the fifty aforementioned film footage representations on scale 24. Each column includes the numerals which will give the correctsolution to a problem wh-ichistobe solved by this side of the instrument. ,As shown in'Figs. 2 and .8, a similar number of columns of numerals, illustrated by .the numeral 2?, is provided on the side of disc 16 adjacent disc 18. The numerals in each column are chosen to give the correct solution to problems which are to be solved by this side .of the instrument. .In the columns oncach side :of the -instrument only re.p-, resentative numerals have been given becauseof the obvious dihiculty of illustrating each .series of, numbers in the fifty columns on each side of the instrument. As will be made clear as the description proceeds, however, a person skilled in the art will :be able to compute the proper numbers to be placed in the columns. It will be noted that circular :dis'c .lifi serves both forms of thefinvention :since one :side thereof .is :used

with the form of the invention shown in Fig. 1 and the other side is used with the form of the invention shown in Fig. 2. In other words, the modifications have a common member. It is obvious that the forms of the invention shown in Figs. 1 and 2 could be as separate instruments but for the animator both sides can be used and therefore they are made into a unitary instrument for his convenience and for the saving of materials.

Circular disc I1, as shown in Figs. 1 and 3, is of a lesser diameter than disc 16, the periphery thereof lying closely adjacent the film footage numerals on disc IS in a manner to permit the viewing of said numerals. Disc I1 is preferably of an opaque material so that the entire columns of numerals on disc I6 will not be visible therethrough. To permit the viewing of particular numerals in the columns, however, a series of spirally arranged rectangular windows 28 is provided. These windows may be in the form of actual openings in the disc, as seen in Fig. 4, or may be transparencies therein. The size of the windows is regulated by the size of the numerals in columns 26. While the windows have been shown to be of rectangular configuration, it is obvious that they could be oval or of any shape which would permit the reading of only one numeral of a series in a particular column. As was noted above, the windows describe a spiral which begins near the mounting pin and extends to the outer periphery of the disc, each window being equally spaced from the windows adjacent thereto. Each window in the series is spaced radially from the mounting pin such a distance that each window will center on or permit the viewing of a separate numeral in the series of numerals in columns 26. That is, the windowv nearest the pivot pin, which begins the spiral, will center on the numerals in the columns nearest to the center pin. The next window in the spiral will read the next numeral outwardly from the aforementioned numeral which is nearest the pivot pin. The succeeding windows will center on succeeding numerals in the column until the last window, which is nearest to the periphery of the disc, centers on the numerals adjacent the periphery of disc l6.

Inscribed on the outer surface of disc I! is a scale 29, said scale being in the shape of an are covering substantially the same number of degrees of are as the spirally arranged windows cover. The scale is marked off with short radial lines, the spacing of which decreases gradually from one end of the scale to the other. The scale is provided with numerals, beginning at one end with A; and ending with 12, each succeeding marking from A; being larger by an amount of The relationship between scale 29 and the spirally arranged windows 28 is such that a straight line may be drawn from the mark on scale 29, through central mounting pin l9, and through the center of the window nearest pin l9. Also, a straight line may be drawn from the 12 mark on scale 29, through the central mounting pin [9. and through the center of the window nearest the periphery of disc [1. Further, a straight line may be drawn from succeeding markings after the A; mark on scale 29, through pin 19, and through succeeding windows after the window nearest pin I9. This relationship between scale 29 and windows 28 is caused by the fact thatas the windows extend spirally outwardly, the actual distance between the windows remains the" same but the angle in degrees between radial linesfrom pin l9 through the center of the windows decreases Therefore, on scale 29, which is an arc with all points equidistant from pin l9, the spacing of the markings gradually decrease from one end to the other. As denoted in the legend to scale 29, the numerals on the scale represent inches, this being the number of inches on the drawing through which the object is to move.

The operation of this side of the instrument which includes disc I! will now be explained in detail. As stated earlier the animator will know the number of feet of film. This is most frequently found by knowing the number of seconds of a narration, during which the movement occurs. By multiplying the number of feet of film used per second by the number of seconds, the footage is obtained. Disc I6 is then rotated until the mark denoting the number of feet obtained above its placed under one end of centerline [4. Then, holding frame member l4 and disc I6 stationary, disc I! is rotated until the mark on scale 29, denoting the number of inches which the object is to move on the drawing, is placed under the other end of centerline l4. With these settings, the answer is shown through the Window 28 which is under the said one end of centerline M. The answer is in hundredths of an inch and tells the operator how much to move the object and pan cel between photographs. The reason for the relationship between which end of centerline I4 is to be placed over the footage scale, the inch scale, and the window showing the answer, is that under the settings aforementioned all the numerals in the problem which are known, and the answer, may be read in an upright position.

From the above it will be apparent that from any known footage over which a known amount of movement in inches is to take place, the animator can determine how much the object should be moved in hundredths of an inch between separate photographs. Since the numerals in columns 25 have not been given in detail the means of computing them will be given. By way of example the column under the footage numeral 1% will be chosen for calculation. By placing numeral 1 /2 under one end of centerline l4 and numeral A; on scale 29 under the other end of said centerline, the window nearest pin l9 will be centered over the lowest numeral position in the column. The numeral to be placed herein may be solved by the following equation:

where With the settings above OMr equals A; inch'or 25 one-hundredths of an inch, and FL equals 1 /2 feet. Substituting these values in Equation 1 the following is obtained:

01W =l.()4l hundredths of an inch This number is rounded off and is placed in the column as 1. To compute the next numeral in the column it is but necessary to use the next numer l in scale 9.. nam y in p ac '01- it used above. Succ ed n numeral i the col ;umn are computed by substitutin succeeding numerals of scale 29 in thecquation. The remaining columns are computed in an identical manner. It will :be noted that .all of the col- :umns vdo not extend the same distance from the periphery toward .pin 19. his is due to .6 fact that the values become so :low as to ,be unusable. That is, the incremental movement of the pan hars is limited to a certain minimum valu the column.

Wh l t e sca d rt n fi m oota e i l mited to 25 -feet, it is obvious that larger footage could. he sought for. For example, if a 'mouement of 6 inches on the drawing is to occur in :50 feet of film it would only be necessary to take one half of each amount and compute this :on the instrument, thus obtaining the correct answer. This is made possible because of the ratio relationship between the numerals on scale '29 and the peripheral footage scale numerals. It will also be apparent that with a known amount of movement per :frame and a known total movement, the footage of f lm required could be idetermined. By placing the total movement numeral on scale ,29 under .one end of lcenterline 1.4 and rotating .disc I6 until the numeral representing the moyement per frame shows up in a window under the other end of said centerline, the footage required will appear .on the footage scale under said other endof the centerline.

To find the total movement on the drawing of the object when the incremental movement .per

frame and the film footage are known, one end 7 viewing of thenumcrals in columns 21 nearest the p ripher an :arcuate scale 32 is :provided on disc 18 on the opposite side of pin 29 from windows 3:]. This scale has all points thereof equidistant from spin 1:9 and is provided with a number .of markings equal to the number :of windows, the spacing of said markings gradually decreasing from one end to the other. The markings on scale .32 are numbered, beginning at the end at which the markings are spaced the furthest apart, as follows: /2,, 1, .2, 3, 4:, v5, 6, '77, :8, -9, 10, and 20. These i-num'bers represent the amount .of object movement on the drawing, in hundredths :of an inch, per frame of film. The relationship "beof the centerllne is placed over the numeral representing the number .of feet of film ,and .disc I! is rotated until the numeral representin the .incremental movement per frame appears through a Window under the same end of the centerline and the total movement in inches may he read on scale 29 under the other end of the centerline.

On the side of the instrument described above, the animator is not concerned with how smooth the object movement will "be, The requisites are that so much movement must take place in a certain number of feet of film. However, there may be instances where the amount of movement per frame ,of film is important to create a special animated e ec e o c. it s necessa y to compute the total amount Of .moyement and thereby the length of background and the ,pan cel.

The other side of the instrument, as shown in Figs. 2 and 8, provides a, means for determining the total movement of an object in hundredths of an inch. As was noted heretofore, the side of disc --I 5 herein involved also has a'pertpherajl footage scale, illustrated by the reference character 30, marked from zero to 25 in equal one-half foot increments. terial and has therein a plurality of spirally arranged windows 3|, beginning at a point midway between the mounting pin is and the periphery of the disc, and extending to the periphery of the disc. As shown in Fig. 5, these windows are actual out-outs in the disc. However, as noted above, the windows may be in the form of transparencies in the opaque disc. The windows are spaced from pin l9 so that the window nearest said pin will show the numerals in columns 2'! nearest the pivot pin. The succeeding windows outwardly will permit the view of outwardly succeed-ing numbers in the columns so that the winde ne r t he disc periphery pe mit. the.

Disc I8 is preferably of opaque matweenscale 3-2 and the spirally arranged windows brought out by the fact that straight lines through pin 19 may be drawn from the /2 mark .on scale ,32 to the window nearest said pin and A from the 20 vmark on scale 32 to the Window furthest from said pin.

With the portion of the invention shown in Figs. 2 and .8 and described above, it is a simple matter to determine the total amount of object movement on the drawing, and thereiore the lengh of the pan -cel upon which the object must be mounted, when the amount of movement per .frame of film, in hundredths of an inch, to create .an animated effect, and the number of feet of film over which the movement is to take place, are known. For example, suppose the animator knows that the amount of movement per frame of fil -m must be hundredth of an inch in order to create a desired effect and that the total movement must be completed in feet of film. The centerline it at one end is placed over numoral 25 .on scale ,30 while the other end of the ,centerline is placed over the numeral A; on scale 32. y reading the numeral exposed by the window beneath the centerline the answer will be found, In this instance, since on scale .32 and Window 3| nearest pin 19 line up in a straight line, the window will permit the viewing of the number on the column extending from the 25 foot mark which is closest to pin 19, the answer well be 200 hundredths of .an inch.

In order that a person skilled in the art could provide the numerals in the various columns illustrated by reference character 2?, .a sample calculation ill be made for the column extending from 162 foot mark on scale til, as was done above in computing the numerals in columns 26. The same Equation 1 is used but is rewritten as follows:

.(2) 0M1:(OM) (FL) (16) By substituting the known information the fol lowing is obtained:

OMt.=.( /2),.(25):(15) =2 00 hundredths of an inch The remaining numerals in the column and the numerals in the other columns are similarly ob tained.

"Disc t8 is also provided with arcuate shaped scales 133 and :34. Scale as is divided into equal size spaces by short radially extending markings and numbered in increments of from '7 through 12. These numbers represent the most often used camera field sizes. Beside numeral 7 is placed the words or below which is self-explanatory. Scale 34 is divided into the same. number of equal spaces as scale 3t? with the legend none placed in the space at one end of the scale and the legend- .any placed in the space at the other end of the scale. Between the two end spaces and begin Hing after the space marked none, the spaces are numbered 6, 9, 12 ;15 2 20, 26 and 34 An unmarked scale 35 is positioned inwardly of scale 34. Scale 35 comprises a'series of short equally spaced radially extending lines, each of said lines being aligned with pivot pin 19 and the center of one of the spaces in scale 34. The numerical value of degrees shown in scale 34 represents various maximum angles from the horizontal which the pan bars may be tilted. The relationships between scales 33, 34 and 35 may be illustrated by the fact that straight lines may be drawn through pin I9 from the numeral 7 on scale 33 to the end marking on scale 35 which is centered on the space in scale 34 having the legend any therein, and from the numeral 12 on scale 33 to the other end marking on scale 35 which is centered on the space in scale 34 having the legend none therein.

Thus, if the animator desires to determine the field size to which the camera must be moved for a given degree of tilt, it is but necessary to line up centerline with the appropriate mark on scale 35 which is centered with the given degree of tilt and the answer is read from scale 33. For example, with the pan bars tilted 12 with respect to the top or bottom of the background drawing, the maximum size field on the drawing which the camera can take in without photographing the pan bars is a 9 field, as shown in Fig. 2. Conversely, with a given field size the animator can determine the maximum number of degrees which the pan bars can be tilted from the horizontal.

Obviously many modifications and variations of the present invention are possible in the light of the above teachings. It is therefore to be understood that within the scope of the appended claims the invention may be practiced otherwise than as specifically described.

I claim:

1. A calculating instrument for solving problems in animated motion picture layout wherein two values are known and a third value is to be solved for comprising a pair of circular discs having centrally arranged apertures therein, an elongated frame means having an aperture centrally arranged therein and a longitudinally extending centerline passing through said aperture, a pin extending through said apertures for mounting said discs on said frame means, one of said discs having a peripheral scale having numerical representations and a plurality of columns of numerals on one side thereof, each column extending radially between the central aperture of said one disc and a numerical representation on said peripheral scale, said columns being angularly spaced from each other by equal amounts, a plurality of spirally arranged windows in the other of said discs, said windows being equally spaced apart and arranged so that succeeding windows in the spiral center on succeeding numerals in said columns of numerals, and an arcuate scale on said other disc having numerical representations thereon, the arrangement being such that when the centerline is placed over the known values in said peripheral scale and said arcuate scale, the value to be solved for may be read through the window under said centerline.

2. A calculating instrument for solving problems in animated motion picture layout wherein two values are known and a third value is unknown, comprising a pair of circular discs having centrally arranged apertures therein, frame means having a pivot pin therein for receiving said apertures to rotatably mount said discs relative to said frame means, said frame means being elongated, and having a longitudinally extending centerline therein, a peripheral scale on one side of one of said discs and having equally spaced graduations thereon, identifying indicia on each of saidgraduations, columns of numerals extending between the identifying indicia and the aperture of said one disc, a plurality of windows arranged in a spiral path in the other of said discs, said windows being arranged so that the window nearest the pivot pin centers on the numerals in said columns nearest said pivot pin and succeeding windows outwardly from said pin center on outwardly succeeding numerals in said columns,

and an arcuate scale having identifying indicia on said other disc, the arrangement of said scales and Windows being such that when one end of the centerline is placed over the graduation in said peripheral scale having an identifying indicia representing one of the known values and the other end of the centerline is placed over the identifying indicia in said arcuate scale which represents the other of said known values, the unknown value may be read from the column of numerals through the window under said one end of the centerline.

3. A calculating instrument for solving problems in animated motion picture layout wherein two values are known and a third value is unknown, comprising a pair of circular discs having centrally arranged apertures therein, frame means having a pivot pin therein for receiving said apertures to rotatably mount said discs relative to said frame means, said frame means being elongated, and having a longitudinally extending centerline therein, a scale having identifying indicia thereon positioned adjacent the periphery of one of said discs, radially extending columns of numerals extending toward the center of said one disc from said identifying indicia, a plurality of equally spaced windows arranged in a spiral path on the other of said discs, said windows being arranged to center on numerals in said columns of numerals, and an arcuate scale on said other disc having identifying indicia thereon, the relative arrangement of said scales and windows being such that by placing said centerline over the identifying indicia in said peripheral scale and said arcuate scale representing said two known values, the unknown value may be read through a window under said centerline from one of said columns of numerals.

4. A calculating instrument as in claim 3 wherein said peripheral scale has equally spaced graduations thereon with which said identifying indicia is associated, and said arcuate scale has graduations thereon which are spaced apart gradually decreasing distances from one end to the other.

5. A calculating instrument as in claim 4 wherein the windows are arranged so that the window nearest the pivot pin centers on the numerals in said columns nearest said pivot pin and succeeding windows center on succeeding numerals in said columns.

6. A calculating instrument as in claim 5 wherein the number of graduations in said arcuate scale is equal to the number of windows.

7. A calculating instrument as in claim 6 wherein the relationship between said spirally arranged windows and said arcuate scale is such that when the longitudinal centerline is placed over one end of the arcuate scale the centerline will overlie the window at one end of the spirally arranged path, and when the centerline is placed over the other end of said scale the centerline team scale from one end to the other may" be placed 3 under said c'enterline, and when so positioned, succeeding windows from one end of said path to the other will be" aligned. with said centerline'.

9'. A calculating instrument as claim 8 wherein the spirally arrang'erf windows are posi- 10 tioned' on one side of said pivot andl sai'd 'arouate scale is positioned on; the" other side thereof.

110". A caIcuIating instnunient as in claim 9' wherein: an additional pair of areuatescales are u nrovidedbn said other disc, said pair of scales havtngx angularly spaced graduations thereon and? i'den'ti' fyingv ind'icia associated. with; each graduation, the graduations in each oi said pair of'scales being equally spaced apart and the gnaduatfons in one of said pair of scalesbeing angularl'y spaced apart the same number of degrees as the graduations in the other oi said pair of scales so that a-known value on said one scale may be placed under the; oenterl-ine and an unknown value will" underlie the centerline onsaid other scale.

No references cited. 

